chap3.txt

  
  3 The new GAP object types of simpcomp
  
  In  order  to meet the particular requirements of piecewise linear geometric
  objects  and  their  invariants, simpcomp defines a number of new GAP object
  types.
  
  All new object types are derived from the object type SCPropertyObject which
  is  a  subtype  of  Record.  It  is a GAP object consisting of permanent and
  temporary  attributes.  While simpcomp makes use of GAP's internal attribute
  caching mechanism for permanent attributes (see below), this is not the case
  for temporary ones.
  
  The temporary properties of a SCPropertyObject can be accessed directly with
  the  functions  SCPropertyTmpByName  and  changed with SCPropertyTmpSet. But
  this  direct  access  to  property  objects is discouraged when working with
  simpcomp,  as  the  internal consistency of the objects cannot be guaranteed
  when the properties of the objects are modified in this way.
  
  Important note: The temporary properties of SCPropertyObject are not used to
  hold  properties  (in  the  GAP  sense)  of  simplicial  complexes  or other
  geometric objects. This is done by the GAP4 type system [BL98]. Instead, the
  properties  handled  by  simpcomp's  own caching mechanism are used to store
  changing  information,  e.g.  the  complex  library  (see Section 13) of the
  package or any other data which possibly is subject to changes (and thus not
  suited to be stored by the GAP type system).
  
  To  realize  its  complex  library  (see Section 13), simpcomp defines a GAP
  object  type  SCLibRepository which provides the possibility to store, load,
  etc.  any  defined geometric object to and from the build-in complex library
  as  well as customized user libraries. In addition, a searching mechanism is
  provided.
  
  Geometric    objects    are    represented    by   the   GAP   object   type
  SCPolyhedralComplex,  which  as  well  is  a  subtype  of  SCPropertyObject.
  SCPolyhedralComplex  is  designed  to represent any kind of piecewise linear
  geometric  object  given  by  a certain cell decomposition. Here, as already
  mentioned,  the  GAP4  type system [BL98] is used to cache properties of the
  object. In this way, a property is not calculated multiple times in case the
  object is not altered (see SCPropertiesDropped (5.1-4) for a way of dropping
  previously calculated properties).
  
  As  of  Version  1.4,  simpcomp  makes  use  of  two  different  subtypes of
  SCPolyhedralComplex:  SCSimplicialComplex to handle simplicial complexes and
  SCNormalSurface to deal with discrete normal surfaces (slicings of dimension
  2). Whenever possible, only one method per operations is implemented to deal
  with  all  subtypes of SCPolyhedralComplex, these functions are described in
  Chapter   4.   For   all   other   operations,  the  different  methods  for
  SCSimplicialComplex and SCNormalSurface are documented separately.
  
  
  3.1 Accessing properties of a SCPolyhedralComplex object
  
  As  described  above  the object type SCPolyhedralComplex (and thus also the
  GAP  object  types  SCSimplicialComplex  and SCNormalSurface) has properties
  that  are  handled  by  the  GAP4  type system. Hence, GAP takes care of the
  internal consistency of objects of type SCSimplicialComplex.
  
  There  are two ways of accessing properties of a SCPolyhedralComplex object.
  The  first is to call a property handler function of the property one wishes
  to  calculate.  The  first  argument  of such a property handler function is
  always  the  simplicial complex for which the property should be calculated,
  in  some  cases  followed  by  further  arguments  of  the  property handler
  function. An example would be:
  
    Example  
    gap> c:=SCBdSimplex(3);; # create a SCSimplicialComplex object
    gap> SCFVector(c);
    [ 4, 6, 4 ]
    gap> SCSkel(c,0);
    [ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ]
  
  
  Here  the functions SCFVector and SCSkel are the property handler functions,
  see Chapter 16 for a list of all property handlers of a SCPolyhedralComplex,
  SCSimplicialComplex  or  SCNormalSurface  object. Apart from this (standard)
  method  of calling the property handlers directly with a SCPolyhedralComplex
  object,  simpcomp provides the user with another more object oriented method
  which calls property handlers of a SCPolyhedralComplex object indirectly and
  more conveniently:
  
    Example  
    gap> c:=SCBdSimplex(3);; # create a SCSimplicialComplex object
    gap> c.F;
    [ 4, 6, 4 ]
    gap> c.Skel(0);
    [ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ]
  
  
  Note  that the code in this example calculates the same properties as in the
  first  example above, but the properties of a SCPolyhedralComplex object are
  accessed via the . operator (the record access operator).
  
  For  each  property  handler  of  a  SCPolyhedralComplex  object  the object
  oriented  form of this property handler equals the name of the corresponding
  operation.  However,  in most cases abbreviations are available: Usually the
  prefix  ``SC''  can  be  dropped,  in  other  cases  even  shorter names are
  available.  See  Chapter  16  for  a  complete  list  of  all  abbreviations
  available.